TIME magazine has been running a series called “Brilliant: The science of smart” by Annie Murphy Paul. The latest column, “Why Guessing Is Undervalued,” quoted several results from research on learning estimation, a topic near to my heart. One result surprised me particularly:
…good estimators possess a clear mental number line — one in which numbers are evenly spaced, or linear, rather than a logarithmic one in which numbers crowd closer together as they get bigger. Most schoolchildren start out with the latter understanding, shedding it as they grow more experienced with numbers.
I do agree that children start out with a logarithmic understanding. I first learned this idea from a wonderful episode of WNYC’s Radio Lab on “Innate numbers” (Nov. 30, 2009). The producers had asked Stanislas Dehaene to discuss his research on innate number perception.
One of his studies involved an Indian tribe in the Amazon. This tribe does not have words for numbers beyond five, and does not have formal teaching of arithmetic.
Lots of readers of my entry on learning languages have said that the only reason I learned French well the second time (with the Assimil course) is that I was motivated. Here is one example: “Guy, the main reason that you learned French this time was because you wanted to learn it this time.”
Understanding the role of motivation in learning is important for designing productive learning environments — i.e. for learning well — so I would like to discuss it further.
Yes, motivation is important for learning! When I was in high school and training for the U.S. Physics Olympiad team, we heard (maybe apocryphal) stories about how our counterparts were being trained in the USSR: Candidates who didn’t make the cut got sent to the army. This kind of motivation, I thought, would definitely lead me to put in the needed hours.
To agree with the readers’ comments more strongly: For learning, motivation is necessary. However, there is a distinction between necessary and sufficient. Although motivation is necessary, it is not sufficient.
My reveries about toasts landing butter side down — the subject of an upcoming blog entry — were pleasantly broken by the discussion on this blog of Ezekiel Emanuel’s analysis of healthcare spending, in which Emanuel goes to town on Left and Right policy ideas for cutting spending. Among his criticisms of Left ideas:
[I]t turns out that the combined profits of the country’s five largest for-profit health insurance companies — United, WellPoint, Aetna, Humana and Cigna — were $11.7 billion, only 0.5 percent of total health-care spending. Even confiscating every penny of those profits would add up to less than half of the cost-saving threshold.
I liked how he quoted the savings as a dimensionless number (the percentage of total healthcare spending).
A few years after I learned German, I got the chance to learn French. That experience gave me lots of ideas for why our teaching of many subjects, especially science and mathematics, is so unsuccessful—and for how we can improve our learning.
I studied French in school for five years. However, when I went to France after college, I could barely buy a train ticket. The impetus to try again came a few years later, in the summer of 1993. Our whole family was going to spend two months in Lyon while my father took a sabbatical. The rest of us enrolled in a four-week language course at the Alliance Française.
While still in America, to get more benefit from the language course, I started relearning French. On the recommendation of a friend who is a linguist and mathematician, I got the self-study French course made by Assimil entitled Le Nouveau Français sans Peine (New French With Ease). (Many other self-study courses should also work well. I have not tried them, so I do not have the knowledge to draw out lessons for learning other subjects, which is my main interest here. But to learn about language programs, I recommend the excellent “How to learn any language” site.)
I did one French lesson daily starting from Lesson 1. I read a short, idiomatic dialogue out loud using the pronunciation key, then listened to it on the tape, repeating it sentence by sentence. The lesson finished with 2 minutes of fill-in-the-word exercises using the vocabulary from the dialogue. Each lesson took about 30 minutes. After three months of this preparation, when I landed in France I could converse with random French people on the train.
I was never good at languages. Although my first language was Punjabi, I grew up as a monolingual English speaker. In grade school, I took French for many years with grades of mostly Bs and a few Cs. However, I managed to learn fairly fluent German in just a few months. As I look back on it, I realize that I applied methods that help in learning any subject, which is my reason for telling you what I did.
It was 20 years ago in the eight-week language course at the Goethe Institute in Prien am Chiemsee, a beautiful resort town in the foothills of the Bavarian Alps (sadly, that school has since closed its doors). Upon arrival, we took placement tests to determine a suitable class. The instructors offered me the choice of starting in the highest of the three beginning levels or in the lowest of the three intermediate levels. (In college I had studied a year of German, which I estimate as comparable to four weeks of immersion in language school.)
I chose the intermediate class. For the first five weeks, I understood almost nothing that the teacher or the other students said. However, in the sixth week of the course, something amazing happened. Each day in that week I understood more.
Did you know that vending machines, not a major danger in most of our minds, are twice as likely to kill you as a shark? I heard this statistic at the new shark-and-ray touch tank of the New England Aquarium, which I try to visit weekly with my daughters. You stand at a large, shallow tank with plexiglass walls and can lay your hand in the water, gently feeling the sharks and sting rays swimming by.
The aquarium probably wants to convince visitors that sharks are not the fierce predators of Jaws fame, and thereby help protect sharks from hunting and extinction. Although I could admire this motive, the comparison always surprised me. My number sense complained that sharks simply must be more dangerous than vending machines.
However, upon looking up the risks, I found that the comparison was correct. The yearly risk (in the United States) of dying from a shark attack is roughly 1 in 250 million. In contrast, the yearly risk of dying from a vending machine accident is roughly 1 in 112 million. The vending machine is indeed roughly twice as lethal as the shark!
Why then was I still troubled by the comparison? Maybe my number sense needed a tune up, and I should just accept the statistical facts of life. I then started thinking about it using the method of easy cases.
In 1776, one book, written in complex language, sold over 120,000 copies in Colonial America. That number does seem large on its own. However, to give it even more meaning, I like to convert it to an equivalent number today.
This conversion is a task for proportional reasoning—one of my favorite tools for finding meaning in the numbers that surround us. First convert 120,000 into a fraction of the U.S. population in 1776: compared to the population at the time of 2.5 million, 120,000 is roughly 1 in 20, or 5%. Today’s U.S. population is about 300 million—of which 5% is 15 million.
Fifteen million copies today! More surprisingly, Common Sense by Thomas Paine sold this equivalent in just three months. In its first year, it sold 500,000 copies, or 20% of the colonial population. Today’s equivalent is 60 million copies. On Wikipedia’s list of bestselling books, all books that have sold that many or more copies have done so over a much longer time. The shortest time is 8 years, for The Da Vinci Code; several others, such as Heidi, were published in the 19th century.
Another surprise arrives upon opening Common Sense: the sophistication of the writing and reasoning.
Our family recently camped for a week in a nearby state forest where our most trusted item was a cast-iron frying pan. Its thickness distributes heat evenly. Nothing can harm it. The wrong kind of spatula won’t scratch some special non-stick coating.With simple care, it will last for a thousand years. Which reminded me how rare that combination of high quality and durability is today.
Most everything else I own is junk and seems to be designed that way. Here are several anecdotal examples:
In the old days, most Americans rented phones from the phone company (“Ma Bell”). My parents still own one, now over 30 years old, that survived raising three boys. These phones lasted forever. Meanwhile, Ma Bell was broken up in the 1980s. One engineer who worked for the phone company before and after the breakup told me of how the engineers were gathered together and given new ground rules: “It was all well and good in the old days to make phones with gold-plated contacts. But now it’s different. Here’s how to make the newer phones…” I think back on this comment as I watch one phone after another die, often after a few months.
Now that the U.S. national debt is in the headlines, the media is awash in astronomical numbers such as $14.3 trillion (the current debt). Everyone realizes that this number is incomprehensible. Even back in 1981, when the national debt was only about $1 trillion, the debt was still incomprehensible. Thus, speechwriters for Ronald Reagan created, or at least popularized, a widely used attempt to give it meaning. Speaking to Congress in February 1981, Reagan said:
A few weeks ago I called such a figure, a trillion dollars, incomprehensible, and I’ve been trying ever since to think of a way to illustrate how big a trillion really is. And the best I could come up with is that if you had a stack of thousand-dollar bills in your hand only 4 inches high, you’d be a millionaire. A trillion dollars would be a stack of thousand-dollar bills 67 miles high.
This comparison is often quoted as a stack of one-dollar bills 67,000 miles high (perhaps because thousand-dollar bills don’t exist). No matter which denomination you use, I give the explanation an A for effort, but an F for performance. For I have little idea of how far 67,000 miles is. I know it’s way too far to walk and even too far to fly (jumbo jets have a maximum range of around 7,000 miles). But is it large as a national debt? I have no idea. Perhaps a large national debt would reach all the way to Mars. The connection to a height has merely replaced one meaningless idea ($1 trillion) with another meaningless idea (a stack 67,000 miles high).
Should professors have tenure? The question, debated recently on this blog, misses the mark—as do the usual answers, whether “yes,” “no,” or “maybe.”
On the “no” side, it is argued that tenure protects incompetent spongers. A very reliable (tenured) colleague, at a university that shall remain nameless, tells me of professors whose interests are no longer intellectual and who spend their time playing the real estate market. Their research productivity, measured in grant dollars or papers, is low; thus, the university is angry. Their teaching is also substandard, yet not quite abysmal enough to get them fired. To urge them to resign, the department punishes them… by assigning extra teaching!
On the “yes” side, it is argued that tenure protects academic freedom. That point is made by my colleague on this blog Dan Hamermesh. Ten years ago I agreed with him. I would not have imagined my future self happy as an associate professor at Olin College of Engineering: Olin offers six-year renewable contracts instead of tenure. Now I see Olin’s system as a reasonable alternative to tenure, for I no longer believe that tenure supports academic freedom.
The “mother of all traffic jams,” in the words of L.A. County supervisor Zev Yaroslavsky, is coming to Los Angeles. On the weekend of July 16/17, an 11-mile segment of Interstate 405 will be closed as part of a $1 billion widening project. Reading of the expected traffic jams, and having recently returned from western Europe, where I traveled mostly by train, I was reminded of an earlier traffic nightmare.
This example I learned from Robert Caro’s 1974 masterpiece The Power Broker: Robert Moses and the Fall of New York. Robert Moses was New York City’s “master builder” in the mid-20th century, and famously hated public transportation.
As a physics student, I found that I could solve most of the problems simply by looking at derivations and listening carefully to my reactions to the equations. A soft voice inside me would say, “No, that term just doesn’t seem right. Go and find out what went wrong there.” Or, “Ah, these terms hang together and the result feels right. It must be okay.” And it almost always worked out. My piano teacher would do the same when playing an unfamiliar piece of music. She could play it just by making sure it sounded right.
Were these just party tricks? Or was a more fundamental process going on?
I was recently traveling in Europe (including in Switzerland, where 87% of trains are less than 3 minutes late). While I was in Cambridge, England, my old friend and colleague David MacKay, who shares my antipathy to bad numbers, gave me a copy of a recent article in the UK journal Physics World (“Optoelectronics: a green explosion”, May 2011, p. 5 of the optics supplement). The article touted laser-based “green technologies,” including their use in reducing carbon-dioxide emissions:
Volvo’s Johnny Larson says it is possible to shave a few kilograms off the weight of a car’s metal frame by optimizing its design for a laser process. This has knocked up to 2 kg [4.4 lbs] off the XC60, and for every one of these models that clocks up 100,000 km [60,000 miles], 24 kg [53 lbs] of carbon-dioxide emissions will be saved.
So many numbers, so little meaning! Whenever I see so many numbers, I think of what Socrates might have said: “The uncompared number is not worth knowing.” Let’s start with the 2-kg weight reduction.
Thanks to recent, hugely popular books about the development of expertise, the term deliberate practice is coming into common usage as the kind of practice that produces expertise.
Deliberate practice requires careful reflection on what worked and what didn’t work. A budding concert pianist may practice a particularly troublesome passage listening for places where his fingers do not flow smoothly. A chess student may spend hours analyzing one move of a world-championship chess match trying to see what the grandmasters saw. This kind of practice demands time for reflection and intense concentration, so intense that it is difficult to sustain for longer than 3 hours per day.
As I have learned more about deliberate practice, I often think about its lessons for the educational system. And they are not happy ones.
It’s been almost a year since the Deepwater Horizon spill in the Gulf of Mexico, and more data is available on the magnitude of the spill. In a recent article, The New York Times describes the spill’s size as “some 4.9 million barrels of oil.” That led me to two numerical musings.
First, I wanted to make the number meaningful by comparing it to another large, famous spill: the Exxon Valdez. Its size is imprinted in my memory as 11 million gallons, which is about 250,000 barrels. So the Deepwater spill was about 20 times larger than the Valdez. In retrospect, that makes sense. The Deepwater well reached into a geologic formation. Whereas the Valdez was merely what one could take from another formation and stuff even into a big ship.
I am fascinated by how we can improve our thinking and problem solving and enjoy learning about and from masters of those arts. My interest was therefore caught by the advice on thinking given in a review of Quantum Man: Richard Feynman’s Life in Science. The reviewer, George Johnson, writes:
This triumph came early in his [Feynman’s] career. His later thinking (about solid-state physics, for example, or quantum cosmology) was just as original. Maybe sometimes too original, Krauss suggests. Science usually proceeds by building on what came before. The maverick in Feynman kept him from accepting even the most established ideas until he had torn them apart and reassembled the pieces. That led to a deeper understanding, but his time might have been better spent at the cutting edge…“He continued to push physics forward as few modern scientists have,” Krauss [the biographer] writes, “but he tended to lead from the rear or, at best, from a side flank.”
Neal Koblitz, professor of mathematics at the University of Washington, begins his critique of computers in K-13 mathematics education as follows: “In Peru, as in many Third World countries, the system of public education is in crisis. Teachers’ pay — traditionally low — is falling rapidly because of inflation. The schools are dilapidated, and there is no money for basic supplies. …”
State educators are becoming increasingly focused on standardized test performance. But does standardized mathematics really teach kids how to reason?
A recent article in The New York Times offers a worrying application of street-fighting reasoning methods. The article describes the deterioration of the Lake Isabella Dam in California. This dam, the article reports, is one of 4,400 considered “susceptible to failure” (out of the 85,000 dams in the country). I’ll pass over lightly the statement early in the article that the repair costs would be “billions of dollars,” and note only that this figure seems like a massive underestimate.
Articles on the health-care industry are a fertile source of large numbers and, sometimes, large errors. It is estimated that nationally 300,000 women a year may be getting unnecessary surgery at a cost of “hundreds of millions of dollars.” I was happy to believe the figure of 300,000 women a year. However, the cost set off my number-sense alarm.
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